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Data Handling & Analysis Allometry & Log-log Regression Andrew Jackson a.jackson@tcd.ie

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Linear type data How are two measures related?

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Length – mass relationships How does the mass of an organism scale with its length? Related to interesting biological and ecological processes – Metabolic costs – Predation or fishing/harvesting – Diet – Ecological scaling laws, fractals and food-web architecture

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Mass of a cube How does a cube scale with its length? – Mass = Density x Volume – Volume = L 1 X L 2 x L 3 = L 3 – Volume = aL b Where a = 1 and b = 3 – So if the cube remains the same shape (i.e. it stays a cube) How does mass change if length is doubled? 2L 1 X 2L 2 x 2L 3 = Mass x 2 3 – Isometic scaling The object does not change shape as is grows or shrinks

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Mass of a sphere How does mass of a sphere change with length? Volume = (4/3)πr 3 = (4/3)π(L 3 /2 3 ) Again, mass changes with Length 3 The difference here, compared with the cube, is the coefficient of Length (4/3)π(L 3 /2 3 ) = (4/3)π(1/2 3 ) (L 3 ) So, we have – Volume = (some number)L 3 – Volume = aL b Where b = 3 in this case

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A general equation Mass = aLength b – Where we might expect b = 3 Take the log of both sides – Log(M) = log(aL b ) – Log(M) = log(a) + log(L b ) – Log(M) = log(a) + b(log(L)) – Y = b 0 + b 1 (X) Log(a) = b 0 So…. a = exp(b 0 ) b 1 = b and is simply the power in the allometric equation

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What do these coefficients mean? On a log-log scale what does the intercept mean? – The intercept is the coefficient, or the multiplier, of length – Mass = aLength b – Spheres and cubes differ only in their coefficients – So a = exp(b 0 ) tells us how the shapes differ between two species

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What does the slope mean? If b1, the slope and coefficient of log(Length) is 3, then the fish grows isometrically – Its shape stays the same

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What do these coefficients mean? If b1, the slope and coefficient of log(Length) is < 3, then the fish becomes thinner as it grows

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What do these coefficients mean? If b1, the slope and coefficient of Length is > 3, then the fish becomes fatter as it grows

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Brain and body mass relationships Instead of plotting brain mass against body mass Plot log(brain mass) against log(body mass)

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Regress brain on body mass Use regression analysis Log(BRAIN) = b 0 + b 1 (Log(BODY)) What value would b 1 take if brains scaled isometrically with body size? A sensible null model would be Brain = aBody 1 i.e. that brain size is a constant proportion of body size In reality, would you expect b 1 to actually be larger or smaller than this? What are the biological reasons that might govern this relationship?

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Common allometric relationships Length scales with Mass 1/3 Surface area scales with Mass 2/3 Metabolic rate scales with Mass 3/4 Breathing rate or Heart rate with Mass 1/4 Abundance of species scales with (Body Mass) 3/4 – except parasites (Hechinger et al 2011, Science, 333, p445-448)

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The slope-intercept form of a linear equation of a non-vertical line is given by: Slope-Intercept Form of a Linear Equation.

The slope-intercept form of a linear equation of a non-vertical line is given by: Slope-Intercept Form of a Linear Equation.

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